Q3) Frequentist and Bayesian estimations [30 Marks] A large manufacturing company, called 3DCotnp, is using several 3D printers to produce spare parts For maehineries and sensors. In order to maintain a reliable production rate for the company. the management of BDComp wants to model the iifetfme of the 3D printers. Through research, 3DComp has found that the income x, (in years) of :1 BD printer, i, follows an exponential distribution with an unknown parameter 6. as shown below. I: ~ Ernie) = ribs-l6) = 994\"\") 3DCornp is interested in nding on average. how fang a certain 3!) primer tests. Since, the value (UH) represents an average, how long a certain 511) printer tests. they are interested in nding an estimate for the parameter 6. 3.1) The company 3DComp first decided to use a frequentist' approach to arrive at an estimate for 9. Answer the following questions. Assume that there are N 3D printers used, and their lifetimes arc independently and identically distributed [iid), a) Show that the expression for the likelihood, p(X|). of N printers {X = {11,x2, , IND can be given by the below equation [Show the steps clearly to obtain this). P0419) = 9N e(Wi' where W = EL 1:, [3 marks] b) Find a simplified expression for the log-likelihood function L(0) = In (p(X|0)) [3 marks] c) Show that the Maximum likelihood Estimate (0 ) of the parameter 0 is given by: N 0 = W [3 Marks] d) From the past history, the company has found out that the lifetimes of eight of the 3D printers were {7, 5, 8, 12, 10, 8, 9, 8}. Find the Maximum likelihood Estimate of the parameter 0 given this data? Hence, find on average, how long a certain 3D printer last? [3 Marks] e) Hence, find the probability that a 3D printer lasts for more than 8 years. (Hint: you may use cumulative distribution function (cdf) of exponential distribution. The cdf of an exponential distribution is given by F(t) = 1 -e-(te)). [4 marks]3.2) The company consulted a 3D printer manufacturing company to learn more about the lifetime of the printers. Through discussions with the engineers of that company, it has found out that the pattern of the parameter {5'} follows a Gamma distribution, Gosh}, as given below. with hyperparameters a = 0.1 and b = 0.2. GUI, b) = Z ba9(a_lje_b3 , where Z is a constant. a) The company, 3DC'omp, decided to use this prior information for their estimation ofthe parameter 3, using Bayesian estimation. I Use the above Gamma distribution as prior [G{a,h)) and obtain an expression for the posterior distribution (show all the steps). I Show that the posterior distribution is also a Gamma distribution, Gta', If), with different hyper-parameters a' and b'. I Express nF and b' in terms of a, b, N and W. [6 Marks] 1)) Using the values a = 0.1 and b = . 2 for the hyper-parameters of the prior, and the data that the company has about the lifetime of the eight 3Dprinters {'F, 5, 8, 12, It], 8, 9, 8}, find the values of a' and b'. What is the posterior mean estimate of 9 1' [4 Marks] e) Write a R program to plot the obtained likelihood distribution, prior distribution, and posterior distribution on the same graph. Use different colors to show the distributions on the plot clearly. [4 Marks]